\(\int \frac {3+e+e^x (9+e^2+36 x+36 x^2+e (6+12 x))}{-27-3 e^2+e (-18-35 x)-105 x-102 x^2+e^x (9+e^2+36 x+36 x^2+e (6+12 x))} \, dx\) [6896]

Optimal result

Integrand size = 76, antiderivative size = 18 \[ \int \frac {3+e+e^x \left (9+e^2+36 x+36 x^2+e (6+12 x)\right )}{-27-3 e^2+e (-18-35 x)-105 x-102 x^2+e^x \left (9+e^2+36 x+36 x^2+e (6+12 x)\right )} \, dx=4+\log \left (-3+e^x+\frac {x}{3+e+6 x}\right ) \]

[Out]

ln(x/(6*x+3+exp(1))+exp(x)-3)+4

Rubi [F]

\[ \int \frac {3+e+e^x \left (9+e^2+36 x+36 x^2+e (6+12 x)\right )}{-27-3 e^2+e (-18-35 x)-105 x-102 x^2+e^x \left (9+e^2+36 x+36 x^2+e (6+12 x)\right )} \, dx=\int \frac {3+e+e^x \left (9+e^2+36 x+36 x^2+e (6+12 x)\right )}{-27-3 e^2+e (-18-35 x)-105 x-102 x^2+e^x \left (9+e^2+36 x+36 x^2+e (6+12 x)\right )} \, dx \]

[In]

Int[(3 + E + E^x*(9 + E^2 + 36*x + 36*x^2 + E*(6 + 12*x)))/(-27 - 3*E^2 + E*(-18 - 35*x) - 105*x - 102*x^2 + E
^x*(9 + E^2 + 36*x + 36*x^2 + E*(6 + 12*x))),x]

[Out]

x - 3*(3 + E)*Defer[Int][(9*(1 + E/3) - 3*(1 + E/3)*E^x + 17*x - 6*E^x*x)^(-1), x] - (3 + E)*Defer[Int][1/((3
+ E + 6*x)*(9*(1 + E/3) - 3*(1 + E/3)*E^x + 17*x - 6*E^x*x)), x] + 17*Defer[Int][x/(-9*(1 + E/3) + 3*(1 + E/3)
*E^x - 17*x + 6*E^x*x), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-3 \left (1+\frac {e}{3}\right )-e^x \left (9+e^2+36 x+36 x^2+e (6+12 x)\right )}{(3+e+6 x) \left (9 \left (1+\frac {e}{3}\right )-3 \left (1+\frac {e}{3}\right ) e^x+17 x-6 e^x x\right )} \, dx \\ & = \int \left (1+\frac {-((3+e) (10+3 e))-35 (3+e) x-102 x^2}{(3+e+6 x) \left (9 \left (1+\frac {e}{3}\right )-3 \left (1+\frac {e}{3}\right ) e^x+17 x-6 e^x x\right )}\right ) \, dx \\ & = x+\int \frac {-((3+e) (10+3 e))-35 (3+e) x-102 x^2}{(3+e+6 x) \left (9 \left (1+\frac {e}{3}\right )-3 \left (1+\frac {e}{3}\right ) e^x+17 x-6 e^x x\right )} \, dx \\ & = x+\int \left (\frac {3 (-3-e)}{9 \left (1+\frac {e}{3}\right )-3 \left (1+\frac {e}{3}\right ) e^x+17 x-6 e^x x}+\frac {-3-e}{(3+e+6 x) \left (9 \left (1+\frac {e}{3}\right )-3 \left (1+\frac {e}{3}\right ) e^x+17 x-6 e^x x\right )}+\frac {17 x}{-9 \left (1+\frac {e}{3}\right )+3 \left (1+\frac {e}{3}\right ) e^x-17 x+6 e^x x}\right ) \, dx \\ & = x+17 \int \frac {x}{-9 \left (1+\frac {e}{3}\right )+3 \left (1+\frac {e}{3}\right ) e^x-17 x+6 e^x x} \, dx+(-3-e) \int \frac {1}{(3+e+6 x) \left (9 \left (1+\frac {e}{3}\right )-3 \left (1+\frac {e}{3}\right ) e^x+17 x-6 e^x x\right )} \, dx-(3 (3+e)) \int \frac {1}{9 \left (1+\frac {e}{3}\right )-3 \left (1+\frac {e}{3}\right ) e^x+17 x-6 e^x x} \, dx \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(37\) vs. \(2(18)=36\).

Time = 5.34 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.06 \[ \int \frac {3+e+e^x \left (9+e^2+36 x+36 x^2+e (6+12 x)\right )}{-27-3 e^2+e (-18-35 x)-105 x-102 x^2+e^x \left (9+e^2+36 x+36 x^2+e (6+12 x)\right )} \, dx=-\log (3+e+6 x)+\log \left (9+3 e-3 e^x-e^{1+x}+17 x-6 e^x x\right ) \]

[In]

Integrate[(3 + E + E^x*(9 + E^2 + 36*x + 36*x^2 + E*(6 + 12*x)))/(-27 - 3*E^2 + E*(-18 - 35*x) - 105*x - 102*x
^2 + E^x*(9 + E^2 + 36*x + 36*x^2 + E*(6 + 12*x))),x]

[Out]

-Log[3 + E + 6*x] + Log[9 + 3*E - 3*E^x - E^(1 + x) + 17*x - 6*E^x*x]

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.39

method	result	size
risch	\(\ln \left ({\mathrm e}^{x}-\frac {3 \,{\mathrm e}+17 x +9}{6 x +3+{\mathrm e}}\right )\)	\(25\)
norman	\(-\ln \left (6 x +3+{\mathrm e}\right )+\ln \left ({\mathrm e} \,{\mathrm e}^{x}+6 \,{\mathrm e}^{x} x -3 \,{\mathrm e}-17 x +3 \,{\mathrm e}^{x}-9\right )\)	\(36\)
parallelrisch	\(-\ln \left (\frac {1}{2}+\frac {{\mathrm e}}{6}+x \right )+\ln \left (\frac {{\mathrm e} \,{\mathrm e}^{x}}{6}+{\mathrm e}^{x} x -\frac {{\mathrm e}}{2}-\frac {17 x}{6}+\frac {{\mathrm e}^{x}}{2}-\frac {3}{2}\right )\)	\(36\)

[In]

int(((exp(1)^2+(12*x+6)*exp(1)+36*x^2+36*x+9)*exp(x)+3+exp(1))/((exp(1)^2+(12*x+6)*exp(1)+36*x^2+36*x+9)*exp(x
)-3*exp(1)^2+(-35*x-18)*exp(1)-102*x^2-105*x-27),x,method=_RETURNVERBOSE)

[Out]

ln(exp(x)-(3*exp(1)+17*x+9)/(6*x+3+exp(1)))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.67 \[ \int \frac {3+e+e^x \left (9+e^2+36 x+36 x^2+e (6+12 x)\right )}{-27-3 e^2+e (-18-35 x)-105 x-102 x^2+e^x \left (9+e^2+36 x+36 x^2+e (6+12 x)\right )} \, dx=\log \left (\frac {{\left (6 \, x + e + 3\right )} e^{x} - 17 \, x - 3 \, e - 9}{6 \, x + e + 3}\right ) \]

[In]

integrate(((exp(1)^2+(12*x+6)*exp(1)+36*x^2+36*x+9)*exp(x)+3+exp(1))/((exp(1)^2+(12*x+6)*exp(1)+36*x^2+36*x+9)
*exp(x)-3*exp(1)^2+(-35*x-18)*exp(1)-102*x^2-105*x-27),x, algorithm="fricas")

[Out]

log(((6*x + e + 3)*e^x - 17*x - 3*e - 9)/(6*x + e + 3))

Sympy [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.33 \[ \int \frac {3+e+e^x \left (9+e^2+36 x+36 x^2+e (6+12 x)\right )}{-27-3 e^2+e (-18-35 x)-105 x-102 x^2+e^x \left (9+e^2+36 x+36 x^2+e (6+12 x)\right )} \, dx=\log {\left (\frac {- 17 x - 9 - 3 e}{6 x + e + 3} + e^{x} \right )} \]

[In]

integrate(((exp(1)**2+(12*x+6)*exp(1)+36*x**2+36*x+9)*exp(x)+3+exp(1))/((exp(1)**2+(12*x+6)*exp(1)+36*x**2+36*
x+9)*exp(x)-3*exp(1)**2+(-35*x-18)*exp(1)-102*x**2-105*x-27),x)

[Out]

log((-17*x - 9 - 3*E)/(6*x + E + 3) + exp(x))

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.67 \[ \int \frac {3+e+e^x \left (9+e^2+36 x+36 x^2+e (6+12 x)\right )}{-27-3 e^2+e (-18-35 x)-105 x-102 x^2+e^x \left (9+e^2+36 x+36 x^2+e (6+12 x)\right )} \, dx=\log \left (\frac {{\left (6 \, x + e + 3\right )} e^{x} - 17 \, x - 3 \, e - 9}{6 \, x + e + 3}\right ) \]

[In]

integrate(((exp(1)^2+(12*x+6)*exp(1)+36*x^2+36*x+9)*exp(x)+3+exp(1))/((exp(1)^2+(12*x+6)*exp(1)+36*x^2+36*x+9)
*exp(x)-3*exp(1)^2+(-35*x-18)*exp(1)-102*x^2-105*x-27),x, algorithm="maxima")

[Out]

log(((6*x + e + 3)*e^x - 17*x - 3*e - 9)/(6*x + e + 3))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.89 \[ \int \frac {3+e+e^x \left (9+e^2+36 x+36 x^2+e (6+12 x)\right )}{-27-3 e^2+e (-18-35 x)-105 x-102 x^2+e^x \left (9+e^2+36 x+36 x^2+e (6+12 x)\right )} \, dx=\log \left (6 \, x e^{x} - 17 \, x - 3 \, e + e^{\left (x + 1\right )} + 3 \, e^{x} - 9\right ) - \log \left (6 \, x + e + 3\right ) \]

[In]

integrate(((exp(1)^2+(12*x+6)*exp(1)+36*x^2+36*x+9)*exp(x)+3+exp(1))/((exp(1)^2+(12*x+6)*exp(1)+36*x^2+36*x+9)
*exp(x)-3*exp(1)^2+(-35*x-18)*exp(1)-102*x^2-105*x-27),x, algorithm="giac")

[Out]

log(6*x*e^x - 17*x - 3*e + e^(x + 1) + 3*e^x - 9) - log(6*x + e + 3)

Mupad [B] (verification not implemented)

Time = 12.65 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.94 \[ \int \frac {3+e+e^x \left (9+e^2+36 x+36 x^2+e (6+12 x)\right )}{-27-3 e^2+e (-18-35 x)-105 x-102 x^2+e^x \left (9+e^2+36 x+36 x^2+e (6+12 x)\right )} \, dx=\ln \left (3\,{\mathrm {e}}^x-3\,\mathrm {e}-17\,x+\mathrm {e}\,{\mathrm {e}}^x+6\,x\,{\mathrm {e}}^x-9\right )-\ln \left (6\,x+\mathrm {e}+3\right ) \]

[In]

int(-(exp(1) + exp(x)*(36*x + exp(2) + 36*x^2 + exp(1)*(12*x + 6) + 9) + 3)/(105*x + 3*exp(2) - exp(x)*(36*x +
exp(2) + 36*x^2 + exp(1)*(12*x + 6) + 9) + 102*x^2 + exp(1)*(35*x + 18) + 27),x)

[Out]

log(3*exp(x) - 3*exp(1) - 17*x + exp(1)*exp(x) + 6*x*exp(x) - 9) - log(6*x + exp(1) + 3)